In a safety squeeze, the runner at third does not take off until the batter makes contact bunting, waiting for more certainty that the ball will go to a location from which it will be difficult for the fielding team to make an out at home plate.
In a suicide squeeze, the runner takes off as soon as the pitcher begins to throw the pitch, before releasing the ball. If properly executed, a play at homeplate is extremely unlikely. However, if the batter fails to make contact with the pitch, the runner is likely to be put out at homeplate (hence, "suicide"). Therefore, the suicide squeeze usually requires a skilled bunter who can make contact consistently, even on poor pitches.
These plays are often used in the late innings of a close game in order to score a pad, winning or tying run.